Speaker
Description
Let $G, H_1,\ldots,H_s \in \mathbb C^{n,n}$ be Hermitian and $S_1,\ldots,S_k \in \mathbb C^{n,n}$ be symmetric matrices. In this talk, we maximize the Rayleigh quotient of a Hermitian matrix $H$ under certain constraints involving Hermitian matrices $H_1,\ldots, H_s$ and symmetric matrices $S_1,\ldots, S_k$. More specifically, we compute
$$
\begin{eqnarray}
& \mu(G,H_1,\ldots,H_s,S_1,\ldots,S_k):=\sup\Big\{\frac{v^*Gv}{v^*v} :~v\in \mathbb C^{n} \setminus \{0\},\,v^*H_iv=0,\, \nonumber \\ &\hspace{6cm} ~i=0,\ldots,s ,\, v^TS_jv=0,~j=1,\ldots,k
\Big\},
\end{eqnarray}
$$
where $T$ and $*$ denote the transpose and the conjugate transpose of a matrix or a vector, respectively. Such problems occur in stability analysis of uncertain systems and in the eigenvalue perturbation theory of matrices and matrix polynomials. The problem without symmetric conditions has been studied by [Bora et al., SIMAX, 35(2), 2014] and a computable formula for $\mu(G,H_1,\ldots,H_s)$ was obtained in terms of the largest eigenvalue of a parameter-dependent Hermitian matrix. Similarly, this problem without Hermitian conditions was considered by [Prajapati & Sharma, LAA, 645, 2022] and a computable formula $\mu(G,S_1,\ldots,S_k)$ was obtained in terms of the second largest eigenvalue of some parameter-dependent Hermitian matrix.
In this talk, we extend the ideas from [Bora et al., SIMAX, 35(2), 2014] and [Prajapati & Sharma, LAA, 645, 2022] and derive a computable estimation for $\mu(G,H_1,\ldots,H_s,S_1,\ldots,S_k)$ with both Hermitian and symmetric constraints.
This estimation is exact when the eigenvalue at the optimal is simple. We then apply these results in computing the structured eigenvalue backward errors of real T-even and real T-odd matrix polynomials. These results are also shown to be useful in obtaining the eigenvalue backward errors of matrix functions involving both Hermitian and skew-symmetric matrices.